Calculating the spectral characteristics of the color resulting from overlaying colorants

A method of operating a processing apparatus to characterize an ink when the ink is printed over a second ink or over an overprint of one or more other inks on a substrate using a printing process, the method comprising:

• (a) accepting in the processing apparatus a spectral measure of reflectance of the substrate, the spectral measure being at a plurality of wavelengths;
• (b) accepting in the processing apparatus the spectral measure of reflectance of the ink printed on the substrate;
• (c) accepting in the processing system the spectral measure of reflectance of a first background printed on the substrate;
• (d) accepting or determining in the processing system the spectral measure of reflectance of the ink printed on the first background;
• and
• (e) determining in the processing system an interaction of absorption and reflection function (“IAR function”) for a plurality of wavelengths indicative of how the ink interacts with a background on which it is printed, and usable for determining the spectral measure of an overprint of the ink over a print of one other ink or an overprint of two or more other inks on the substrate by multiplying the spectral measure of reflectance of the print or overprint of one or more other inks by the exponentiation of
• (i) the ratio of the spectral measure of the ink on the substrate to the spectral measure of the substrate, by
• (ii) the TAR function of the ink.

A non-transitory computer-readable medium with a set of instructions thereon that when executed by one or more processors of a processing system cause carrying out a method of determining a spectral measure of reflectance of an overprint of a plurality of inks made on a substrate using a printing process, the method comprising:

• (a) accepting in the processing system a spectral measure of reflectance of a substrate at a plurality of wavelengths;
• (b) accepting in the processing system a pre-defined order of printing an overprint of a plurality of inks on the substrate;
• (c) for each ink of the plurality of inks, for a respective amount of printing of the ink, accepting or determining in the processing system a respective interaction of absorption and reflection function (“IAR function”) indicative of how the ink interacts with a layer of one or more other inks when printed over the layer of the one or more other inks;
• and
• (d) determining in the processing system the spectral measure of reflectance of the overprint of the plurality of inks printed on the substrate in the pre-defined order in the respective amounts of the inks, including repeatedly multiplying, in the pre-defined order, for each additional ink added to a partial overprint of the inks before the additional ink is added, the spectral measure of the partial overprint by the exponentiation of:
• (i) the ratio of the spectral measure of the additional ink on the substrate to the spectral measure of the substrate
• by
• (ii) the TAR function of the additional ink,
• starting with spectral measure of the substrate.

An apparatus to determine a spectral measure of reflectance of an overprint of a plurality of inks made on a substrate using a printing process, the apparatus comprising:

• one or more processors;
• and
• storage,
• wherein the storage comprising instructions that when carried out by one or more of the apparatus to:
• (a) accept in the apparatus a spectral measure of reflectance of a substrate at a plurality of wavelengths;
• (b) accept a pre-defined order of printing an overprint of a plurality of inks on the substrate;
• (c) for each ink of the plurality of inks, for a respective amount of printing of the ink, accept or determine a respective interaction of absorption and reflection function (“IAR function”) indicative of how the ink interacts with a layer of one or more other inks when printed over the layer of the one or more other inks;
• and
• (d) determine the spectral measure of reflectance of the overprint of the plurality of inks printed on the substrate in the pre-defined order in the respective amounts of the inks, including repeatedly multiplying, in the pre-defined order, for each additional ink added to a partial overprint of the inks before the additional ink is added, the spectral measure of the partial overprint by the exponentiation of:
• (i) the ratio of the spectral measure of the additional ink on the substrate to the spectral measure of the substrate,
• by
• (ii) the TAR function of the additional ink,
• starting with spectral measure of the substrate.